smc

The area of a triangle is $\frac{1}{2}bh.$ To maximize the base, let it be equal to the diameter of the semi circle, which is equal to $2r.$ To maximize the height, or altitude, choose the point directly in the middle of the arc connecting the endpoints of the diameter. It is equal to $r.$ Therefore the area is $\frac{1}{2}\cdot 2r\cdot r=\boxed{r^2}$.